Many important properties are identified and criteria are developed for the existence of subquasigroups in finite quasigroups. Based on these results, we propose an effective method that concludes the nonexistence of proper subquasigroups in a given finite quasigroup, or finds all its proper subquasigroups. This has an important application in checking the cryptographic suitability of a quasigroup. Using arithmetic of finite fields, we introduce a binary operation to construct quasigroups of order p r . Criteria are developed under which the quasigroups mentioned have desirable cryptographic properties, such as polynomial completeness and absence of proper subquasigroups. Effective methods are given for constructing cryptographically suitable quasigroups. The efficiency of the methods is illustrated by some academic examples and implementation of all proposed algorithms in the computer algebra system Singular .

It is proved that a ring $A$ is a right or left Noetherian, right distributive centrally essential ring if and only if $A=A_1\times\cdots\times A_n$, where each of the rings $A_i$ is either a commutative Dedekind domain or a uniserial Artinian centrally essential (not necessarily commutative) ring. V.T.Markov is supported by the Russian Foundation for Basic Research, project 17-01-00895-A. A.A.Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013.

In this paper, we obtain a classification of quasigroup rings by the quantity of elements with null left annihilator for different quasigroups. This classification becomes possible due to a criterion of being an element with null left annihilator in a quasigroup ring. By virtue of this criterion, we make a calculation to find regularities using various fields and quasigroups with order 4. This outcome helps us to obtain two results where any two quasigroup rings have the same number of elements with null left annihilator and the element of the quasigroup ring GF(p)Q with fixed quasigroup Q has null left annihilator in the quasigroup ring GF(pn)Q.

In this paper, we identify many important properties and develop criteria for the existence of subquasigroups in finite quasigroups. Based on these results, we propose an effective method that concludes the nonexistence of subquasigroup of a finite quasigroup, otherwise finds its all possible proper subquasigroups. This has an important application in checking the cryptographic suitability of a finite quasigroup. \par Further, we propose a binary operation using arithmetic of finite fields to construct quasigroups of order $p^r$. We develop the criteria under which these quasigroups have desirable cryptographic properties, viz. polynomially completeness and possessing no proper subquasigroups. Then a practical method is given to construct cryptographically suitable quasigroups. We also illustrate these methods by some academic examples and implement all proposed algorithms in the computer algebra system {\sc{Singular}}.

In this review, we consider applications of nonassociative algebraic structures for the construction of linearly optimal codes and cryptosystems.

It is proved that a ring $R$ is a right uniserial, right Noetherian centrally essential ring if and only if $R$ is a commutative discrete valuation domain or a left and right Artinian, left and right uniserial ring. It is also proved that there exist non-commutative uniserial Artinian centrally essential rings. Victor Markov is supported by the Russian Foundation for Basic Research, project 17-01-00895-A. Askar Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013.

Large order cryptographically suitable quasigroups have important applications in the development of crypto-primitives and cryptographic schemes. These present new perspectives of cryptography and information security. From algebraic point of view polynomial completeness is one of the most important characteristic for cryptographically suitable quasigroups. In this paper, we propose four different methods to construct polynomially complete quasigroups of any order [Formula: see text]. First method is based on a starting quasigroup of same order, second method is based on a particular permutation of [Formula: see text] and third and fourth methods are based on products of lower order quasigroups. In the last case, all quasigroups and their isotopes are polynomially complete. We also develop and implement an algorithm to derive a permutation for a given permutation of [Formula: see text] so that they generate whole [Formula: see text].

Mathematical objects in this paper are group codes. In the first part of the paper we present a survey with some of the main results about group codes, mainly the existence of group codes that are not abelian group codes, the minimal length and the minimal dimension of such codes and the existence of a non-abelian group code that has better parameters than any abelian group code. In particular, in a previous paper [1], we have shown that the minimal dimension of a group code that is not abelian group code is 4. However, all known examples of group codes of dimension 4 that are non-abelian group codes are constructed using groups that are not p-groups. We do not know if such codes exist for the case of p-groups, but in the second part of this paper we prove that, under some restrictions on the base field, all four-dimensional G-codes for an arbitrary finite p-group G are abelian.

It is proved that for any prime integer p and each field F of characteristic p, there exists a centrally essential F-algebra which is not a PI-ring and is not algebraic over its center.

A ring R with center C is said to be centrally essential if the module R-C is an essential extension of the module C-C. We describe centrally essential exterior algebras of finitely generated free modules over not necessary commutative rings and study properties of semi-Artinian centrally essential rings.

Centrally essential rings were defined earlier for associative unital rings; in this paper, we define them for rings which are not necessarily associative or unital. In this case, it is proved that centrally essential semiprime rings are commutative. It is proved that all idempotents of a centrally essential alternative ring are central. Several examples of non-commutative centrally essential rings are provided, some properties of centrally essential rings are described.

It is proved that a finite-dimensional centrally essential algebra R is a centrally essential ring if and only if the formal power series ring is centrally essential, if and only if the formal Laurent series ring is centrally essential. There exists a centrally essential ring R such that the ring with respect to the Jacobson radical is not a PI ring; consequently, the ring with respect to the prime radical also is not a PI ring (in particular, the rings and are not commutative).

We describe associative center [Formula: see text] and the center [Formula: see text] of the ring [Formula: see text] obtained by applying the generalized Cayley–Dickson construction and we find conditions under which the ring [Formula: see text] is [Formula: see text]-essential or centrally essential. The obtained results are applied to generalized quaternion rings and octonion rings; we use them to construct an example of a nonassociative centrally essential ring.

A ring R with center C is said to be centrally essential if the module RC is an essential extension of the module CC. In the paper, we study groups whose group algebras over fields are centrally essential rings. We focus on the centrally essential modular group algebras of finite groups over fields of nonzero characteristic.

Let F be a finite field and let G be a finite group. We show that if C is a G-code over F with dimF⁡(C)≤3 then C is an abelian group code. Since there exist non-abelian group codes of dimension 4 when charF>2 (see the examples in [1]), we conclude that the smallest dimension of a non-abelian group code over a finite field is 4.

We prove that there exist non-Abelian group codes over an arbitrary finite field.

We analyze algorithms for open construction of a key on some noncommutative group. Algorithms of factorization and decomposition for associative algebras (of small dimension) are considered. A survey of applications (in particular, in cryptography) of so-called “hidden matrices” is given.

We investigate the possibility to use non-associative groupoids in the realization of an open key distribution procedure based on a generalization of the well known Diffie-Hellman algorithm. We prove the existence of non-associative groupoids which are simultaneously power commuting and not power-associative.

It has been known some time ago that there are one-sided group codes that are not abelian codes, however the similar question for group codes was not known until we constructed an example of a non-abelian group code using the group ring F5S4. The proof needs some computational help, since we need to know the weight distribution of all abelian codes of length 24 over the prime field of 5 elements. It is natural to ask, is it really relevant that the group ring is semisimple? What happens in the case of characteristic 2 and 3? Our interest to these questions is connected also with the following open question: does the property of all group codes for the given group to be abelian depend on the choice of the base field (the similar property for left group codes does)? We have addressed this question, again with computer help, proving that there are also examples of non-abelian group codes in the non-semisimple case. The results show some interesting differences between the cases of characteristic 2 and 3. Moreover, using the group SL(2, F-3) instead of the symmetric group we can prove, without using a computer for it, that there is a code over F-2 of length 24, dimension 6 and minimal weight 10. It has greater minimum distance than any abelian group code having the same length and dimension over F-2, and moreover this code has the greatest minimum distance among all binary linear codes with the same length and dimension. The existence of such code gives a good reason to study non-abelian group codes.